section 3.4 properties of logarithmsmrsk.ca/ap/propertiesoflogs.pdf · section 3.4 – properties...
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27
Section 3.4 – Properties of Logarithms
Product Rule
log log logb b b
MN M N
For M > 0 and N > 0
log
log
b
b
x
y
M x M b
N y N b
MN = bx b
y = b
x+y
Convert back to logarithmic form: logb
MN x y
log log logb b b
MN M N
Example
Use the product rule to expand the logarithmic expression
log(100 ) log100 logx x
Power Rule
log log
b bM M
pp
Example
Use the power rule to expand each logarithmic expression
1/33ln ln( )x x 13
ln x
Quotient Rule
log log log b b b
MN
M N
Example
Use the quotient rule to expand the logarithmic expression
5 511
ln ln ln11 5 ln11e e
28
Express each of the following in terms of sums and differences of logarithms
a) 6
log (7.9)
6 6 6
log ( . ) l7 7g9 o 9log Product Rule
b) 9log 15
7
9 9 9log log15 15 7
7log Quotient Rule
c) 5
log 8
5 5
1/2log lo 8g8
5
12
8log Power Rule
d) 4 43 3log log logx y x yb b b
Product Rule
= 1/34log logx yb b
Power Rule
= 4 xb
log + 3
1 y
blog
e) 2 42 4
log log loga a a
p rmnq
mnp r
q
Quotient Rule
2 4log log log log loga a a a a
m n q p r Product Rule
2 4log log log log loga a a a a
m n q p r
log log log 2log 4loga a a a a
m n q p r Power Rule
f) 33
1/ 2log log log 255 5 525
xx y
y
Quotient Rule
2 31/ 2log log 5 log5 5 5
x y Product Rule
= 5
2 31/ 2log log 5 log5 5
x y
2og 5 2
5l
1
log 2 3log5 52
x y
29
Example
Write as a single logarithmic
a) log25 log4 log (25).(4)
log100
2log10
2
b) log( ) log lo 7 67 6 gxx
xx
c) 3 3 3 3 3
2log log log l( 2) ( ) 2og o2 l gx x x x Product Rule
3
( 2l g
)
2o
x x
Quotient Rule
d) 2 1/313
2ln ln( 5) ln ln( 5)x x x x Power Rule
2 1/3ln ( 5)x x Product Rule
2 3ln 5x x
e) 22log( 3) log log( 3) logx x x x
Power Rule
2( 3)
logx
x
Quotient Rule
f) 1/4 2 101
log 2log 5 10log log log 5 log4
x y x yb b b b b b
Power Rule
1/4 2 10log log 5 logx y
b b b
Factor the minus
1/4 2 10log log 5x y
b b
Product Rule
4
2 105log x
yb
Quotient Rule
30
Exercises Section 3.4 – Properties of Logarithms
1. Express as a sum of logarithms: 3
log ( )ab
2. Express as a sum of logarithms: 7
(7 )log x
3. Express the following in terms of sums and differences of logarithms 1000
log x
4. Express the following in terms of sums and differences of logarithms 5
125logy
5. Express the following in terms of sums and differences of logarithms 7log xb
6. Express the following in terms of sums and differences of logarithms 7ln x
7. Express the following in terms of sums and differences of logarithms 2
log4
x y
a z
8. Express the following in terms of sums and differences of logarithms 2
3log
b
x y
b
9. Express the following in terms of sums and differences of logarithms 3
2log
x y
b z
10. Express the following in terms of sums and differences of logarithms
5
43log
z
yx
b
11. Express the following in terms of sums and differences of logarithms 3 3
2
100 5log
3( 7)
x x
x
12. Express the following in terms of sums and differences of logarithms 4 8 12
3 5
log m n
a a b
13. Express the following in terms of sums and differences of logarithms 5 4
32
logp
m n
t
14. Express the following in terms of sums and differences of logarithms 3 5
logb
nm
x y
z
15. Express the following in terms of sums and differences of logarithms 2
5 log a b
a c
3
31
16. Express the following in terms of sums and differences of logarithms 43log x yb
17. Express the following in terms of sums and differences of logarithms
3255log
y
x
18. Express 3
2 4log
ax w
y z in terms of logarithms of x, y, z, and w.
19. Express 43
loga
y
x z in terms of logarithms of x, y, and z.
20. Express 7
45
ln x
y z in terms of logarithms of x, y, and z.
21. Express 4
35
lny
xz
in terms of logarithms of x, y, and z.
22. Express as one logarithm: 13
2log log 2 5log 2 3a a a
x x x
23. Express as one logarithm: 12
5log log 3 4 3log 5 1a a a
x x x
24. Express as one logarithm: 3 2 3log 2log 3log xx y x yy
25. Express as one logarithm: 3 3 613
ln ln 5lny x y y
26. Express as one logarithm: 12ln 4ln 3lnx xyy
27. Write the expression as a single logarithm. 4ln 7ln 3lnx y z
28. Write the expression as a single logarithm. 213
5ln( 6) ln ln( 25)x x x
29. Write the expression as a single logarithm. 22 ln 4 ln 2 ln( )3
x x x y
30. Write the expression as a single logarithm. 2312 2
log log 2 logb b b
m n m n
31. Write the expression as a single logarithm. 3 4 4 31 22 3
log logy y
p q p q
32. Write the expression as a single logarithm. xyxaaa logloglog 34
21
33. Write the expression as a single logarithm. 223
ln 9 ln 3 lnx x x y
32
34. Write the expression as a single logarithm. 1
log 2log 5 10log4
x yb b b
35. Assume that10
log 2 .3010 . Find each logarithm 10
log 4 , 10
log 5
36. Given that: log 2 0.301a
, log 7 0.845a
, and log 11 1.041a
find each of the following:
2 1log , log 14, log 98, log , log 911 7a a a a a
35
Solution Section 3.4 – Properties of Logarithms
Exercise
Express as a sum of logarithms: 3
log ( )ab
Solution
3 3 3log ( ) log logab a b
Exercise
Express as a sum of logarithms: 7
(7 )log x
Solution
7 7 7(7 ) 7log log logx x
71 log x
Exercise
Express the following in terms of sums and differences of logarithms 1000
log x
Solution
1000loglog1000
log xx
Exercise
Express the following in terms of sums and differences of logarithms 5
125logy
Solution
5 5 5125125log log log y
y
Exercise
Express the following in terms of sums and differences of logarithms 7log x
b
Solution
xb
xb
log77log
36
Exercise
Express the following in terms of sums and differences of logarithms 7ln x
Solution
1/77ln ln xx
17
ln x
Exercise
Express the following in terms of sums and differences of logarithms 2
log4
x y
a z
Solution
2 2 4log log log4
x yx y z
a a az Quotient Rule
2 4
log log logx y za a a
Product Rule
2log log 4logx y za a a
Power Rule
Exercise
Express the following in terms of sums and differences of logarithms 2
3log
b
x y
b
Solution
32
3
2
logloglog byxb
yx
bbb
32
logloglog byxbbb
byxbbb log3loglog2
32 loglog yxbb
37
Exercise
Express the following in terms of sums and differences of logarithms 3
2log
x y
b z
Solution
3
3 22
log log logx y
x y zb b bz
3 2log log logx y zb b b
3log log 2logx y zb b b
Exercise
Express the following in terms of sums and differences of logarithms
5
43log
z
yx
b
Solution
43
4 535
log log logx y
x y zb b bz
= 3/1log xb
+ 4log yb
- 5log zb
= 3
1 x
blog + 4 y
blog - 5 z
blog
Exercise
Express the following in terms of sums and differences of logarithms 3 3
2
100 5log
3( 7)
x x
x
Solution
3 3
3 232
100 5log log 100 5 log 3( 7)
3( 7)
x xx x x
x
1/32 3 2log10 log log 5 log3 log ( 7)x x x
13
2log10 3log log 5 log3 2log( 7)x x x
13
2 3log log 5 log3 2log( 7)x x x
38
Exercise
Express the following in terms of sums and differences of logarithms 4 8 12
3 5
log m n
a a b
Solution
8 12
3 5
1/44 8 12
log3 5
log m na
a b
m n
a a b
Power Rule
= 41 8 12
3 5log m n
aa b
Quotient Rule
8 12 3 51 log log4 a am n a b
Product Rule
8 12 3 51 log log
4log loga am n a aa b
Power Rule
8log log 3 5log1 12
4 a a am n b
3 52log 3log log4 4a a am n b
Exercise
Use the properties of logarithms to rewrite: 5 4
32
logp
m n
t
Solution
1/35 4 5 4
32 2
log logp p
m n m n
t t
Power Rule
5 4
21 log3 p
m n
t
Quotient Rule
5 4 21 log log3 p p
m n t
Product Rule
5 4 21 log log log3 p p p
m n t
Power Rule
1 5log 4log 2log3 p p p
m n t
5 4 2log log log3 3 3p p p
m n t
39
Exercise
Express the following in terms of sums and differences of logarithms 3 5
logb
nm
x y
z
Solution
1/3 5 3 5
log logb b
n
nm m
x y x y
z z
3 5
1 logb m
x y
n z
Power Rule
3 51 log logb b
mx y zn
Quotient Rule
3 51 log log logb b b
mx y zn
Product Rule
1 3log 5log logb b b
x y m zn
Power Rule
3 5log log logb b b
mn n n
x y z
Exercise
Express the following in terms of sums and differences of logarithms 2
5 log a b
a c
3
Solution
1 / 32 2
5 5 log log
a aa b a b
c c
3
Convert the radical to power
2
5
1 log3 a
a b
c
Power Rule
2 51 log log3 a a
a b c
Quotient Rule
2 51 log log log3 a a a
a b c
Product Rule
1 2log log 5log3 a a a
a b c
Power Rule
52 1log log log3 3 3a a a
a b c
52 1 log log3 3 3a a
b c
40
Exercise
Express the following in terms of sums and differences of logarithms 43log x yb
Solution
43log x yb
= 4 3log logx yb b
= 4log xb
+ 1/ 3log yb
= 4 log xb
+ 3
1 log y
b
Exercise
Express the following in terms of sums and differences of logarithms
3255log
y
x
Solution
33
1 2 255 5 525
x /log log x log yy
=
2/1
5log x - 32
5log5
5log y
=
2/1log
5x - 25
5log - 3
5log y
= 2
1 x5
log - 2 55
log - 3 y5
log
= 2
1 x5
log - 2 - 3 y5
log
Exercise
Express 3
2 4log
ax w
y z in terms of logarithms of x, y, z, and w.
Solution
3 3 2 42 4
log log loga a a
x w x w y zy z
Quotient rule
3 2 4log log log loga a a a
x w y z
Product rule
41
3 2 4log log log loga a a a
x w y z
Distribute minus
3log log 2log 4loga a a a
x w y z
Power rule
Exercise
Express 43
loga
y
x z in terms of logarithms of x, y, and z.
Solution
1/2 4 1/34 3
log log loga a a
yy x z
x z
Quotient rule
1/2 4 1/3log log log
a a ay x z
Product rule
1/2 4 1/3log log loga a a
y x z
Distribute minus
1 1log 4log log2 3a a a
y x z
Power rule
Exercise
Express 7
45
ln x
y z in terms of logarithms of x, y, and z.
Solution
1/47 7
45 5
ln lnx x
y z y z
7
51 ln4
x
y z
Power rule
7 51 ln ln4
x y z
Quotient rule
7 51 ln ln ln4
x y z
Product rule
7 51 ln ln ln4
x y z
1 7 ln 5ln ln4
x y z
Power rule
7 5ln ln ln4 4
x y z
42
Exercise
Express 4
35
ln y
xz
in terms of logarithms of x, y, and z.
Solution
1/34 4
35 5
ln ln lny y
x xz z
Product rule
4/3
5/3ln ln
yx
z
4/3 5/3ln ln lnx y z
Quotient rule
54ln ln ln3 3
x y z
Power rule
Exercise
Express as one logarithm: 13
2log log 2 5log 2 3a a a
x x x
Solution
1/3 5213
2log log 2 5log 2 3 log log 2 log 2 3a a a a a a
x x x x x x
1/3 52log 2 log 2 3
a ax x x
1/32 2
52 3
logx x
ax
Exercise
Express as one logarithm: 12
5log log 3 4 3log 5 1a a a
x x x
Solution
1/2 3512
5log log 3 4 3log 5 1 log log 3 4 log 5 1a a a a a a
x x x x x x
1/2 35log log 3 4 log 5 1
a a ax x x
1/2 35log log 3 4 5 1
a ax x x
5
1/2 33 4 5 1
log xa
x x
43
Exercise
Express as one logarithm: 3 2 3log 2log 3log xx y x yy
Solution
2 3
3 2 3 2 1/3 13log 2 log 3log log log logxx y x y x y xy xyy
3 2 2 2/3 3 3log log logx y x y x y
3 2 2 2/3 3 3log logx y x y x y
3 2 5 7/3log logx y x y
3 2
5 7/3log
x y
x y
2 7/3
2log
y y
x
13/3
2log
y
x
3 13
2log
y
x
43
2log
y y
x
Exercise
Express as one logarithm: 3 3 613
ln ln 5lny x y y
Solution
1/3
3 3 6 3 3 6 513
ln ln 5ln ln ln lny x y y y x y y
3 3/3 6/3 5ln ln lny x y y
3 2 5ln ln lny xy y
3 2 5ln lny xy y
5
5ln
y x
y
ln x
44
Exercise
Express as one logarithm: 12ln 4ln 3lnx xyy
Solution
4
321 12ln 4 ln 3ln ln ln lnx xy x xyy y
2 4 3 3ln ln lnx y x y
2 4 3 3ln lnx y x y
2 1 3ln lnx y x
2
1 3ln x
y x
ln
y
x
Exercise
Write as a single logarithmic 4ln 7ln 3lnx y z
Solution
4 7 34ln 7ln 3ln ln ln lnx y z x y z
4 7 3ln lnx y z
4 7
3ln
x y
z
Exercise
Write as a single logarithmic 213
5ln( 6) ln ln( 25)x x x
Solution
213
5ln( 6) ln ln( 25)x x x 21
35ln( 6) ln ln( 25)x x x
5 213
ln( 6) ln ( 25)x x x
5
2
( 6)13 ( 25)
lnx
x x
5
2
1/3( 6)
( 25)ln
x
x x
45
Exercise
Write as a single logarithmic 22 ln 4 ln 2 ln( )3
x x x y
Solution
22 42 2ln 4 ln 2 ln( ) ln ln( )
3 3 2xx x x y x yx
( 2)( 2)2 ln ln( )3 2
x xx y
x
2 ln( 2) ln( )3
x x y
2/3ln( 2) ln( )x x y
2/3ln( 2) ( )x x y
3 2ln( ) ( 2)x y x
Exercise
Write as a single logarithmic 2312 2
log log 2 logb b b
m n m n
Solution
3/22 1/2 231
2 2log log 2 log log log 2 log
b b b b b bm n m n m n m n
3/21/2 2log 2 logb b
m n m n
1/2 3/2 3/2
22log
bm n
m n
3/2 1/2
3/22log
bn
m
3
3
1/22log
bn
m
3
8logb
n
m
46
Exercise
Write the expression as a single logarithm. 3 4 4 31 22 3
log logy y
p q p q
Solution
1/2 2/3
3 4 4 3 3 4 4 31 22 3
log log log logy y y y
p q p q p q p q
1/23 4
2/34 3
logy
p q
p q
1/2 1/23 4
2/3 2/34 3
logy
p q
p q
3/2 2
8/3 2log
y
p q
p q
3/2
8/3log
y
p
p
8/3 3/21log
yp
7/61log
yp
Exercise
Write the expression as a single logarithm. xyxaaa logloglog 34
21
Solution
xyxyxaaaaa logloglogloglog 2
521 434
2/54loglog xy
aa
54loglog xy
aa
47
Exercise
Write the expression as a single logarithm. 223
ln 9 ln 3 lnx x x y
Solution
222 2 ln
3 39ln 9 ln 3 ln ln
3xx x x y x yx
2 ln3
3 ( 3)ln
3
x xx y
x
= 3
2ln( 3)x + ln(x + y)
= 2/3
ln 3 ln( )x x y
= 2/3ln 3 ( )x x y
= 23ln ( ) 3x y x
Exercise
Write the expression as a single logarithm. 1
log 2log 5 10log4
x yb b b
Solution
1log 2log 5 10log
4x y
b b b =
1/4 2 10log log 5 logx yb b b
1/4 2 10log log 5 logx y
b b b
1/4 2 10log log 5x y
b b
4
2 105log x
yb
Exercise
Assume that10
log 2 .3010 . Find each logarithm 10
log 4 , 10
log 5
Solution
a) 10 10
2log 4 log 2
10
2log 2
2(.301)
.6020
48
b) 10 10
102
log 5 log
10 10
log 10 log 2
1 .03010
.6990
Exercise
Given that: log 2 0.301a
, log 7 0.845a
, and log 11 1.041a
find each of the following:
2 1log , log 14, log 98, log , log 911 7a a a a a
Solution
2log log 2 log 1111a a a
0.301 1.041 0.74
log 14 log 2(7) log 2 log 7 0.301 0.845 1.146a a a a
2 2log 98 log 2(7 ) log 2 log 7 log 2 2log 7 0.301 2(0.845) 1.991
a a a a a a
1log log 1 log 7 0 0.845 0.8457a a a
log 9a
Can’t be found from the given information